Relativ-ist-ic Topics. heh. Based on Einstein's "The Electrodynamics of Moving Bodies" which contained his special theory of relativity (1905).
A. Time Dilation (p66)
- where v represents the speed of a moving object relative to the stationary object, t is the time interval as witnessed by the stationary observer and t' is the time interval according to the person in motion. (Can you be sure who is really in motion?)
- If v is small, relative to the speed of light, the denominator approaches 1, so the perceived time difference between the 2 points of reference (time dilation) is close to 0.
- As v approaches c, the denominator of the fraction approaches 0, thus time dilation increases toward infinity.
- Each observer sees his own clock as moving normally.
- Any clock moving relative to the observer will seem to the observer to be running slowly.
- c (speed of light) is the same from all points of view, .
B. Relativistic Addition of Velocities (p110)
- At relatively slow speeds, if a person in a vehicle that is moving toward you throws a ball toward you, the ball appears to be moving very fast. In other words it appears that , where u is the apparent velocity of the ball from your point of view, u' is the velocity at which he throws it at you, and v is his velocity relative to you. (Again, does it matter who is really moving? Can you even tell?) To simplify, it looks like the ball is moving at his velocity PLUS the thrown velocity.
- Einstein modified the equation ...
- If v and u' are small, the part in parenthesis approaches 0, thus the denominator approaches 1, and the answer is similar to the answer from the classical equation.
- As v and u' approach the speed of light, u approaches the speed of light as well. Here's where Einstein realized/proved that u cannot exceed the speed of light. Try the equation using larger and larger numbers (but not larger than c). See?
C. The Equivalence of Mass and Energy (p190)
- Traditionally Kinetic Energy, the energy an object has because of it's mass and motion is calculated as:
- Einstein derived a new equation based on the principles of special relativity: , where v is the velocity of the object.
- Note the extra added to the end of the equation so that KE=0 when v=0. This has nothing to do with the velocity of the object. At this point Einstein had discovered that, even at rest, an object has some amount of energy simply as a result of having mass. An object's rest energy is proportional to its mass:
- This huge amount of energy hidden in even the smallest masses is easier to visualize when you realize how LITTLE mass is required to make a nuclear weapon, or try to fathom the incredible amount of energy given off by the Sun because of these same nuclear forces.
- This energy/mass relationship relates to the universe as a while. "The total energy of the universe equals the total mass of the universe times the speed of light squared.
- This also means that mass can be converted into energy (and back) as in the afore mentioned nuclear reactions.
- This also makes it easier to accept the basic premise of the "Big Bang" theory. Perhaps all of the mass of the universe WASN'T squashed into a softball sized thing 13 billion years ago. Maybe it wasn't mass at all. Maybe it was ALL energy.
D. Orbiting Satellites and Black Holes (p266)
- , where M is the mass of the large object that you are trying to escape, R is its radius, and G is Newton's universal gravitational constant
- Find the escape velocity of the Earth if and,
- What would its escape velocity be if we squeezed all of the Earth into a sphere with a radius of 1m?
- What if we were dealing with an object 1000 times more massive than the Earth, with a 1m radius?
- Our Sun has a mass 333,000 times the mass of the Earth. What would its escape velocity be if it was only 1m radius?
- What happens when large stars die? Much of their mass gets squeezed into small spheres. If their escape velocity is greater than , what does that mean?
E. Quantum Angular Momentum (p302)
- Electron Spin
F. DeBroglie Waves (p466)
- , where h is Planck's constant,
- The dual nature of light...
G. The Doppler Effect and the Big Bang (p504)
H. Electron Tunneling (p714)